The NCTM (2000) data analysis and probability standards state that students should "develop and evaluate inferences and predictions that are based on data." In grades 6-8 classrooms, students are expected to develop and evaluate inferences and predictions in order to do the following:

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Use observations about differences between two or more samples to make conjectures about the populations from which the samples were taken

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Make conjectures about possible relationships between two characteristics of a sample on the basis of scatter plots of the data and approximate lines of fit

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Use conjectures to formulate new questions and plan new studies to answer them

Inference and prediction are more advanced aspects of working with data, as they require an understanding of sampling. Students in grades 6-8 are developing an understanding of the idea of sampling. They often still expect their own intuition to be more reliable than the information they are obtaining from the data. Students begin to develop an understanding of these statistical ideas through conversations as they consider what the data are telling us, what might account for these results, and whether this would be true in other similar situations. Students' early experiences are often with census data -- e.g., the population of their class. When they begin to wonder what might be true for other classes and other schools, they begin to develop the skills of inference and prediction. In the later middle grades and in high school, students begin to learn ways of quantifying how certain one can be about statistical results.

When viewing the video segment, keep the following questions in mind:

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How does Mr. Sowden encourage students to make inferences and predictions?

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What are some of the students' preliminary conclusions?

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Which of the students' inferences are based on the data, and which are based on their own personal judgement?

Video SegmentIn this video segment, Paul Sowden asks the students to look for patterns in the four line plots and to try to determine where the coins with no marking were minted. Students discuss the variance in the data and speculate on why the coins have no mint marks and about where those coins might have been minted.

If you're using a VCR, you can find this segment on the session video approximately 48 minutes and 33 seconds after the Annenberg Media logo.

Problem C1

Answer the questions you reflected on as you watched the video:

a.

How does Mr. Sowden encourage students to make inferences and predictions?

b.

What are some of the students' preliminary conclusions?

c.

Which of the students' inferences are based on the data, and which are based on their own personal judgement?

Problem C2

In the video segment from Part A, students considered why this set of coins contained more coins from Philadelphia. One student hypothesized that this was because Philadelphia is the closest U.S. Mint. This student was beginning to think about the sample of coins the students were using. How might you facilitate a discussion with your students about bias in data and the extent to which a data set can be representative? What questions would you pose? What issues would you raise?

Problem C3

If you were teaching this lesson on investigating nickels and their mint marks, what questions might you ask to focus students' attention on each of the following central elements of statistical analysis: Note 5

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Defining the population

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Defining an appropriate sample

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Collecting data from that sample

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Describing the sample

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Making reasonable inferences relating the sample and the population

Join the discussion! Post your answer to Problem C3 on Channel Talk, then read and respond to answers posted by others.

Problem C4

In the video, the students used circle graphs and line plots to examine the variation in their data. The teacher plans to continue analyzing the variation in the data, using different types of representations. What other types of representations might he use to examine the data?

Problem C5

How could you extend the discussion in this video segment to bring out more speculations about the nickels? How might you formalize these notions into stated conjectures that could be investigated further? What are some conjectures that might arise? How could you formulate them into new questions? How could these questions then be investigated?

Join the discussion! Post your answer to Problem C5 on Channel Talk, then read and respond to answers posted by others.